Random simple connected graph generation with given sparseness

Question

I'm trying to find an efficient algorithm to generate a simple connected graph with given sparseness. Something like:

Input:
    N - size of generated graph
    S - sparseness (numer of edges actually; from N-1 to N(N-1)/2)
Output:
    simple connected graph G(v,e) with N vertices and S edges

Answer 1

High-Level Idea

1. Generate a (uniformly chosen) random spanning tree with N nodes and N - 1 edges.
2. Until the requested number of edges has been reached, add an edge between any two random nodes.

Creating the Spanning Tree

The partition-based answer by ypnos is a good start, but bias is introduced by always selecting a visited node for one end of a new edge. By randomly selecting a visited node at each iteration, nodes that are visited towards the beginning have more iterations from which they have a chance to be chosen. Therefore, earlier nodes are more likely to have a high degree (number of edges) than those picked later.

Example of Bias

As an example, for a 4 node connected graph rather than generating a linear path graph, which is what 75% of the possible spanning trees are, this kind of bias will cause the star graph to be generated with greater than the 25% probability that it should be.

Bias isn't always a bad thing. It turns out this kind of bias is good for generating spanning trees that are similar to real world computer networks. However, in order to create a truly random connected graph the initial spanning tree must be picked uniformly from the set of possible spanning trees (see Wikipedia's Uniform Spanning Tree article).

Random Walk Approach

One approach to generating a uniform spanning tree is through a random walk. Below is a quote from the paper Generating Random Spanning Trees More Quickly than the Cover Time by Wilson describing simple random walk algorithm.

Start at any vertex and do a simple random walk on the graph. Each time a vertex is first encountered, mark the edge from which it was discovered. When all the vertices are discovered, the marked edges form a random spanning tree. This algorithm is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities.

This works well for a simple connected graph, however if you need an algorithm for a directed graph then read the paper further as it describes Wilson's Algorithm. Here is another resource for random spanning trees and Wilson's Algorithm.

Implementation

As I was also interested in this problem, I coded Python implementations of various approaches, including the random walk approach. Feel free to take a look at the Gist of the code on GitHub.

Below is an excerpt from the code of the random walk approach:

# Create two partitions, S and T. Initially store all nodes in S.
S, T = set(nodes), set()

# Pick a random node, and mark it as visited and the current node.
current_node = random.sample(S, 1).pop()
S.remove(current_node)
T.add(current_node)

graph = Graph(nodes)

# Create a random connected graph.
while S:
    # Randomly pick the next node from the neighbors of the current node.
    # As we are generating a connected graph, we assume a complete graph.
    neighbor_node = random.sample(nodes, 1).pop()
    # If the new node hasn't been visited, add the edge from current to new.
    if neighbor_node not in T:
        edge = (current_node, neighbor_node)
        graph.add_edge(edge)
        S.remove(neighbor_node)
        T.add(neighbor_node)
    # Set the new node as the current node.
    current_node = neighbor_node

# Add random edges until the number of desired edges is reached.
graph.add_random_edges(num_edges)

Answer 2

For each node you need at least one edge.

Start with one node. In each iteration, create a new node and a new edge. The edge is to connect the new node with a random node from the previous node set.

After all nodes are created, create random edges until S is fulfilled. Make sure not to create double edges (for this you can use an adjacency matrix).

Random graph is done in O(S).

Answer 3

Based on Wesley Baugh's answer I came up with the following javascript implementation with cytoscape.js to handle graphs:

function generateRandomGraph(cy, numNode, avgDegree, weightMin, weightMax) {
  // create nodes
  for (var i = 0; i < numNode; i++) {
    cy.add({
      group: "nodes",
      data: {
        id: "n" + i
      }
    });
  }

  // perform random walks to connect edges
  var nodes = cy.nodes(),
    S = nodes.toArray(),
    T = []; // visited

  var currNodeIdx = randomIntBetween(0, S.length);
  var currNode = S[currNodeIdx];
  S.splice(currNodeIdx, 1);
  T.push(currNode);

  while (S.length > 0) {
    var neighbourNodeIdx = randomIntBetween(0, S.length);
    var neighbourNode = S[neighbourNodeIdx];
    cy.add({
      group: "edges",
      data: {
        weight: randomIntBetweenInclusive(weightMin, weightMax),
        source: currNode.id(),
        target: neighbourNode.id()
      }
    });
    S.splice(neighbourNodeIdx, 1);
    T.push(neighbourNode);
    currNode = neighbourNode;
  }

  // add random edges until avgDegree is satisfied
  while (nodes.totalDegree() / nodes.length < avgDegree) {
    var temp = sampleInPlace(nodes, 2);
    if (temp[0].edgesWith(temp[1]).length === 0) {
      cy.add({
        group: "edges",
        data: {
          weight: randomIntBetweenInclusive(weightMin, weightMax),
          source: temp[0].id(),
          target: temp[1].id()
        }
      })
    }
  }
}

generateRandomGraph(cy, 20, 2.8, 1, 20);

For complete example source code, visit my github repo :)